# Extended Echelon Form and Four Subspaces

Robert A. Beezer

Department of Mathematics and Computer Science

University of Puget Sound

beezer@pugetsound.edu

Revised: October 23, 2013; November 5, 2013

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Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank.

# AcknowledgementsAcknowledgements

Jeff Stuart provided a literature search of similar results in introductory textbooks, of which there was only one. Helpful suggestions from a referee have improved this note, and are greatly appreciated.